Class Test covering
\nRolling a pair of dice. Find probability that at least one die shows a given number.
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "Two fair six-sided dice are rolled.
", "advice": "\n \n \nLet $A$ be the event that first dice shows a $\\var{number}$ $\\Rightarrow P(A)=\\frac{1}{6}$.
\n \n \n \nLet $B$ be the event that second dice shows a $\\var{number}$ $\\Rightarrow P(B)=\\frac{1}{6}$.
\n \n \n \n$A$ and $B$ are independent events so $P(A\\cap B) = P(A)\\times P(B)$.
\n \n \n \nWe want the probability $P(A \\cup B)$ of either $A$ or $B$ showing $\\var{number}$ and
\n \n \n \n\\[\\begin{eqnarray*}\n \n P(A \\cup B) &=& P(A)+P(B)-P(A \\cap B)\\\\\n \n &=& P(A)+P(B)-P(A)P(B)\\\\\n \n &=&\\frac{1}{6}+ \\frac{1}{6}-\\frac{1}{36}\\\\\n \n &=& \\frac{11}{36}\n \n \\end{eqnarray*}\n \n \\]
\n \n \n ", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "variables": {"number": {"name": "number", "group": "Ungrouped variables", "definition": "random(1..6)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["number"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "What is the probability of at least one die showing a $\\var{number}$?
\nProbability = [[0]]
", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "11/36", "maxValue": "11/36", "correctAnswerFraction": false, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Cillian's copy of Probability: Elementary probability", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ruth Hand", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3228/"}, {"name": "Cillian Williamson", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/10206/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "A bag contains:
$\\var{srn}$ small, red tokens,
$\\var{sbn}$ small, blue tokens,
$\\var{brn}$ large, red tokens, and
$\\var{bbn}$ large, blue tokens.
A probability is a fraction. You can give your answer as a fraction, decimal or percentage as these are all equivalent.
The formula for probability is:
\\[ P(A) = \\frac{\\text{number of possibilities for A}}{\\text{number of total possible outcomes}} \\]
\nFor this question the total possible outcomes are $\\var{srn}+\\var{sbn}+\\var{brn}+\\var{bbn} = \\var{total}$.
Therefore
\\[ P(\\text{A large red token}) = \\frac{\\var{brn}}{\\var{total}} = \\var[fractionnumbers]{brn/total}\\]
\nFor this question we need to know the total number of small tokens, i.e. $\\var{srn}+\\var{sbn} = \\var{srn+sbn}$.
Therefore
\\[ P(\\text{A small token}) = \\frac{\\var{srn+sbn}}{\\var{total}} = \\var[fractionnumbers]{(srn+sbn)/total}\\]
", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"srn": {"name": "srn", "group": "Ungrouped variables", "definition": "random(1..20)", "description": "", "templateType": "anything", "can_override": false}, "brn": {"name": "brn", "group": "Ungrouped variables", "definition": "random(1..20)", "description": "", "templateType": "anything", "can_override": false}, "sbn": {"name": "sbn", "group": "Ungrouped variables", "definition": "random(1..20)", "description": "", "templateType": "anything", "can_override": false}, "bbn": {"name": "bbn", "group": "Ungrouped variables", "definition": "random(1..20)", "description": "", "templateType": "anything", "can_override": false}, "total": {"name": "total", "group": "Ungrouped variables", "definition": "brn+bbn+srn+sbn", "description": "", "templateType": "anything", "can_override": false}, "ans1": {"name": "ans1", "group": "Ungrouped variables", "definition": "precround(brn/total,2)", "description": "", "templateType": "anything", "can_override": false}, "ans2": {"name": "ans2", "group": "Ungrouped variables", "definition": "precround((srn+sbn)/total,2)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["srn", "brn", "sbn", "bbn", "total", "ans1", "ans2"], "variable_groups": [{"name": "Unnamed group", "variables": []}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "You take a token at random.
What is the probability that it is a large, red token?
Give your answer as a fraction, or a decimal correct to 2dp.
You take a token at random.
What is the probability that it is a small token?
Give your answer as a fraction, or a decimal correct to 2dp.
Calculate the probability that none of the $\\var{n}$ students in the sample cycle to college.
", "correctAnswerStyle": "plain", "minValue": "(q^n)-0.001", "variableReplacementStrategy": "originalfirst"}, {"scripts": {}, "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "marks": "5", "variableReplacements": [], "type": "numberentry", "allowFractions": false, "showFeedbackIcon": true, "maxValue": "answer2 +0.001", "correctAnswerFraction": false, "prompt": "Calculate the probability that at least $\\var{r}$ of the $\\var{n}$ students cycle to college.
", "correctAnswerStyle": "plain", "minValue": "answer2 -0.001", "variableReplacementStrategy": "originalfirst"}], "rulesets": {}, "tags": ["Binomial", "binomial", "REBEL", "Rebel", "rebel", "rebelmaths"], "variables": {"pr1": {"name": "pr1", "group": "Ungrouped variables", "definition": "n*p*q^(n-1)", "templateType": "anything", "description": "probability that r = 1
"}, "r0": {"name": "r0", "group": "Ungrouped variables", "definition": "0", "templateType": "anything", "description": ""}, "p_perc": {"name": "p_perc", "group": "Ungrouped variables", "definition": "p*100", "templateType": "anything", "description": "percentage of students that cycle to college
"}, "pr2": {"name": "pr2", "group": "Ungrouped variables", "definition": "((n*(n-1))/2)*(p^2)*q^(n-2)", "templateType": "anything", "description": "probability that r = 2
"}, "q": {"name": "q", "group": "Ungrouped variables", "definition": "1-p", "templateType": "anything", "description": "probability tha an individual does not cycle to college
"}, "answer2": {"name": "answer2", "group": "Ungrouped variables", "definition": "1-answer1", "templateType": "anything", "description": ""}, "pr3": {"name": "pr3", "group": "Ungrouped variables", "definition": "((n*(n-1)*(n-2))/6)*(p^3)*(q^(n-3))", "templateType": "anything", "description": "probability that r = 3
"}, "r": {"name": "r", "group": "Ungrouped variables", "definition": "3", "templateType": "anything", "description": "more than r of the students cycle to college
"}, "n2": {"name": "n2", "group": "Ungrouped variables", "definition": "n-2", "templateType": "anything", "description": ""}, "answer1": {"name": "answer1", "group": "Ungrouped variables", "definition": "if(r=2,pr0+pr1, pr0+pr1+pr2)", "templateType": "anything", "description": ""}, "p": {"name": "p", "group": "Ungrouped variables", "definition": "random(0.1..0.4#0.05)", "templateType": "anything", "description": "the probability that an individual student cycles to college
"}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "random(6..12)", "templateType": "anything", "description": "sample size
"}, "qn": {"name": "qn", "group": "Ungrouped variables", "definition": "q^n", "templateType": "anything", "description": ""}, "pr0": {"name": "pr0", "group": "Ungrouped variables", "definition": "q^n", "templateType": "anything", "description": "probability that r = 0
"}}, "functions": {}, "advice": "Part (a)
\nIf a random variable $X$ follows a binomial distribution with parameters $n$ and $p$. The probability of $r$ successes out of $n$ trials is given by:
\n$P(X=r)=P(r,n)=C^n_{r}p^{r}q^{n-r}$
\nwhere $p$ is the probability of success for each trial and $q$ is the probability of failure for each trial.
\nThe probability that a student cycles to college is $\\var{p}$, therefore $p=\\var{p}$ and $q=1-\\var{p}=\\var{q}$.
\nWe are interested in claculating the probability that none of the sample of $\\var{n}$ students cycle to college so $r=0$ and $n=\\var{n}$
\n$P(\\var{r0}, \\var{n})= C^\\var{n}_{\\var{r0}}$ $\\var{p}^\\var{r0}$ $\\var{q}^{\\var{n}-\\var{r0}}$
\n$P(\\var{r0}, \\var{n})= \\var{pr0}$
\n\n
Part (b)
\nWe are interested in claculating the probability that at least $\\var{r}$ of the $\\var{n}$ students cycle to college. Let $X$ represent the number of students that cycle to college. We need to calculate:
\n$P(X \\geq \\var{r}) = P(X= \\var{r}) + P(X= \\var{r+1})+...+ P(X=\\var{n})$
\n\n
Since $P(X=\\var{r0})+P(X=\\var{r0+1})+...+P(X=\\var{n})=\\var{r0+1}$
\nWe may write
\n$P(X \\geq \\var{r}) = 1-P(X= \\var{r0}) - P(X=\\var{r0+1})-...- P(X=\\var{r-1})$
\n\n
where
\n$P(X= \\var{r0})=P(\\var{r0}, \\var{n})= C^\\var{n}_{\\var{r0}}$ $\\var{p}^\\var{r0}$ $\\var{q}^{\\var{n}-\\var{r0}}=\\var{pr0}$
\n$P(X=1) =P(1, \\var{n})= C^\\var{n}_{1}$ $\\var{p}^{1}$ $\\var{q}^{\\var{n}-1}$ $=\\var{pr1}$
\n$P(X=2) = P(2, \\var{n})=$ $C^\\var{n}_{2}$ $\\var{p}^{2}$ $\\var{q}^{\\var{n}-2}$ $=\\var{pr2}$
\n\nThen
\n$P(X \\geq \\var{r}) = 1-\\var{qn}-\\var{pr1}-\\var{pr2}=\\var{answer2}$
", "metadata": {"description": "It is estimated that 30% of all CIT students cycle to college. If a random sample of eight CIT students is chosen, calculate the probability that...
\nrebelmaths
", "licence": "Creative Commons Attribution 4.0 International"}, "ungrouped_variables": ["p", "p_perc", "n", "q", "r", "pr0", "pr1", "pr2", "pr3", "answer1", "answer2", "qn", "r0", "n2"], "variablesTest": {"condition": "", "maxRuns": 100}, "variable_groups": [], "statement": "Please give your answer to at least 3 decimal places.
\nIt is estimated that $\\var{p_perc}$% of all CIT students cycle to college. A random sample of $\\var{n}$ CIT students is chosen.
\n", "type": "question"}, {"name": "Clodagh's copy of BS3.3 Binomial Cookies", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Clodagh Carroll", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/384/"}, {"name": "Catherine Palmer", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/423/"}], "tags": ["Binomial Distribution", "binomial distribution", "Binomial distribution", "expectation", "expected number", "probabilities", "probability", "Probability", "rebelmaths", "sc", "standard deviation", "statistical distributions", "statistics"], "metadata": {"description": "rebelmaths
\nApplication of the binomial distribution given probabilities of success of an event.
\nFinding probabilities using the binomial distribution.
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "{pre} $\\var{thismany}${post}
\n{something} $\\var{number1}$ {else}
\n", "advice": "
a)
\n$X \\sim \\operatorname{bin}(\\var{number1},\\var{prob})$, so $n= \\var{number1},\\;\\;p=\\var{prob}$.
\n\nb)
\n1. \\[ \\begin{eqnarray*}\\operatorname{P}(X = \\var{thisnumber}) &=& \\dbinom{\\var{number1}}{\\var{thisnumber}}\\times\\var{prob}^{\\var{thisnumber}}\\times(1-\\var{prob})^{\\var{number1-thisnumber}}\\\\& =& \\var{comb(number1,thisnumber)} \\times\\var{prob}^{\\var{thisnumber}}\\times\\var{1-prob}^{\\var{number1-thisnumber}}\\\\&=&\\var{prob1}\\end{eqnarray*} \\] to 3 decimal places.
\n\n
2.
\n\\[ \\begin{eqnarray*}\\operatorname{P}(X \\leq \\var{thatnumber})& =& \\simplify[all,!collectNumbers]{P(X = 0) + P(X = 1) + {v}*P(X = 2)+ {v1}*P(X = 3)}\\\\& =& \\simplify[zeroFactor,zeroTerm,unitFactor]{{1 -prob} ^ {number1}+ {number1} *{prob} *{1 -prob} ^ {number1 -1} + {v} * ({number1} * {number1 -1}/2)* {prob} ^ 2 *( {1 -prob} ^ {number1 -2})+ {v1} * ({number1} * {number1 -1}*{number1-2}/(3*2))* {prob} ^ 3 *( {1 -prob} ^ {number1 -3})}\\\\& =& \\var{prob3}\\end{eqnarray*} \\]
\nto 3 decimal places.
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"templateType": "anything", "can_override": false}, "tol": {"name": "tol", "group": "Ungrouped variables", "definition": "0.001", "description": "", "templateType": "anything", "can_override": false}, "prob": {"name": "prob", "group": "Ungrouped variables", "definition": "thismany/100", "description": "", "templateType": "anything", "can_override": false}, "thisaswell": {"name": "thisaswell", "group": "Ungrouped variables", "definition": "\"our selection contains no more than \"", "description": "", "templateType": "anything", "can_override": false}, "else": {"name": "else", "group": "Ungrouped variables", "definition": "\"biscuits are selected at random.\"", "description": "", "templateType": "anything", "can_override": false}, "v1": {"name": "v1", "group": "Ungrouped variables", "definition": "if(thatnumber=3,1,0)", "description": "", "templateType": "anything", "can_override": false}, "number1": {"name": "number1", "group": "Ungrouped variables", "definition": 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"thismany": {"name": "thismany", "group": "Ungrouped variables", "definition": "random(8..43)", "description": "", "templateType": "anything", "can_override": false}, "this": {"name": "this", "group": "Ungrouped variables", "definition": "\"our selection contains exactly \"", "description": "", "templateType": "anything", "can_override": false}, "v": {"name": "v", "group": "Ungrouped variables", "definition": "if(thatnumber=1,0,1)", "description": "", "templateType": "anything", "can_override": false}, "tprob1": {"name": "tprob1", "group": "Ungrouped variables", "definition": "comb(number1,thisnumber)*prob^thisnumber*(1-prob)^(number1-thisnumber)", "description": "", "templateType": "anything", "can_override": false}, "tprob3": {"name": "tprob3", "group": "Ungrouped variables", "definition": "tprob3a+tprob3b+tprob3c", "description": "", "templateType": "anything", "can_override": false}, "tprob2": {"name": "tprob2", "group": "Ungrouped variables", "definition": "if(thatnumber=2,(1-prob)^number1+number1*prob*(1-prob)^(number1-1)+number1*(number1-1)*prob^2*(1-prob)^(number1-2)/2,(1-prob)^number1+number1*prob*(1-prob)^(number1-1))", "description": "", "templateType": "anything", "can_override": false}, "sd": {"name": "sd", "group": "Ungrouped variables", "definition": "precround(sqrt(number1*prob*(1-prob)),3)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["pre", "thatnumber", "this", "things", "prob1", "descx", "descx1", "thisnumber", "else", "thismany", "number1", "something", "tol", "v", "tprob1", "post", "tprob2", "prob2", "prob", "thisaswell", "sd", "v1", "tprob3a", "prob3", "tprob3b", "tprob3c", "tprob3"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": 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Assuming a binomial distribution for {descX}, write down the values of $n$ and $p$.
\n$n=\\; $[[0]] $p=\\;$[[1]]
\n\n", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "number1", "maxValue": "number1", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "prob", "maxValue": "prob", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Find the probability that {this} $\\var{thisnumber}$ {things}
\n$\\operatorname{P}(r=\\var{thisnumber})=$ [[0]] (to 3 decimal places).
\n\n
Find the probability that {thisaswell} {thatnumber} {things}
\n$\\operatorname{P}(r\\leq\\var{thatnumber})=$ [[1]] (to 3 decimal places).
", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "3", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "prob1-tol", "maxValue": "prob1+tol", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "3", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "prob3-tol", "maxValue": "prob3+tol", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}]}, {"name": "Normal Distribution", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["", ""], "variable_overrides": [[], []], "questions": [{"name": "Cillian's copy of Probabilities from a normal distribution - Electrician", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}, {"name": "Cillian Williamson", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/10206/"}], "preamble": {"css": "", "js": ""}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "Given a random variable $X$ normally distributed as $\\operatorname{N}(m,\\sigma^2)$ find probabilities $P(X \\gt a),\\; a \\gt m;\\;\\;P(X \\lt b),\\;b \\lt m$.
\nrebelmaths
"}, "ungrouped_variables": ["units1", "upper", "lower", "p1", "m", "s", "zupper", "p", "amount", "p2", "tol", "zlower", "stuff", "prob2", "prob3", "prob1"], "parts": [{"showCorrectAnswer": true, "type": "gapfill", "marks": 0, "showFeedbackIcon": true, "gaps": [{"type": "numberentry", "marks": "4", "showFeedbackIcon": true, "correctAnswerStyle": "plain", "minValue": "prob2-tol", "variableReplacementStrategy": "originalfirst", "allowFractions": false, "showCorrectAnswer": true, "variableReplacements": [], "correctAnswerFraction": false, "maxValue": "prob2+tol", "notationStyles": ["plain", "en", "si-en"], "scripts": {}}], "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "scripts": {}, "prompt": "If one electrician is chosen at random, what is probability that this electrician earns over €{upper}?
\nProbability = [[0]](to 2 decimal places)
\n"}, {"showCorrectAnswer": true, "type": "gapfill", "marks": 0, "showFeedbackIcon": true, "gaps": [{"type": "numberentry", "marks": "4", "showFeedbackIcon": true, "correctAnswerStyle": "plain", "minValue": "prob1-tol", "variableReplacementStrategy": "originalfirst", "allowFractions": false, "showCorrectAnswer": true, "variableReplacements": [], "correctAnswerFraction": false, "maxValue": "prob1+tol", "notationStyles": ["plain", "en", "si-en"], "scripts": {}}], "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "scripts": {}, "prompt": "If one electrician is chosen at random, what is probability that this electrician earns less than €{lower} ?
\nProbability = [[0]](to 2 decimal places)
"}, {"showCorrectAnswer": true, "type": "gapfill", "marks": 0, "showFeedbackIcon": true, "gaps": [{"type": "numberentry", "marks": "4", "showFeedbackIcon": true, "correctAnswerStyle": "plain", "minValue": "prob3-tol", "variableReplacementStrategy": "originalfirst", "allowFractions": false, "showCorrectAnswer": true, "variableReplacements": [], "correctAnswerFraction": false, "maxValue": "prob3+tol", "notationStyles": ["plain", "en", "si-en"], "scripts": {}}], "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "scripts": {}, "prompt": "If one electrician is chosen at random, what is probability that this electrician earns between €{lower} and €{upper}?
\nProbability = [[0]](to 2 decimal places)
"}], "functions": {}, "variable_groups": [], "variables": {"zupper": {"name": "zupper", "description": "", "definition": "precround((upper-m)/s,2)", "templateType": "anything", "group": "Ungrouped variables"}, "p2": {"name": "p2", "description": "", "definition": "1-p", "templateType": "anything", "group": "Ungrouped variables"}, "amount": {"name": "amount", "description": "", "definition": "\"electricity consumption\"", "templateType": "anything", "group": "Ungrouped variables"}, "units1": {"name": "units1", "description": "", "definition": "\"k Wh\"", "templateType": "anything", "group": "Ungrouped variables"}, "upper": {"name": "upper", "description": "", "definition": "random(m+0.5s..m+1.5*s#10000)", "templateType": "anything", "group": "Ungrouped variables"}, "prob3": {"name": "prob3", "description": "", "definition": "precround(p1-p2,2)", "templateType": "anything", "group": "Ungrouped variables"}, "s": {"name": "s", "description": "", "definition": "random(5000..15000#1000)", "templateType": "anything", "group": "Ungrouped variables"}, "prob1": {"name": "prob1", "description": "", "definition": "precround(1-p,2)", "templateType": "anything", "group": "Ungrouped variables"}, "m": {"name": "m", "description": "", "definition": "random(30000..50000#2000)", "templateType": "anything", "group": "Ungrouped variables"}, "prob2": {"name": "prob2", "description": "", "definition": "precround(1-p1,2)", "templateType": "anything", "group": "Ungrouped variables"}, "tol": {"name": "tol", "description": "", "definition": "0.01", "templateType": "anything", "group": "Ungrouped variables"}, "lower": {"name": "lower", "description": "", "definition": "random(m-1.5*s..m-0.5s#10000)", "templateType": "anything", "group": "Ungrouped variables"}, "stuff": {"name": "stuff", "description": "", "definition": "\"a frozen foods warehouse each week in the summer months \"", "templateType": "anything", "group": "Ungrouped variables"}, "zlower": {"name": "zlower", "description": "", "definition": "precround((m-lower)/s,2)", "templateType": "anything", "group": "Ungrouped variables"}, "p1": {"name": "p1", "description": "", "definition": "precround(normalcdf(zupper,0,1),4)", "templateType": "anything", "group": "Ungrouped variables"}, "p": {"name": "p", "description": "", "definition": "precround(normalcdf(zlower,0,1),4)", "templateType": "anything", "group": "Ungrouped variables"}}, "variablesTest": {"maxRuns": 100, "condition": ""}, "rulesets": {}, "advice": "1. Converting to $\\operatorname{N}(0,1)$
\n$\\simplify[all,!collectNumbers]{P(X > {upper}) = P(Z > ({upper} -{m}) / {s})} = P(Z>\\var{zupper}) = 1-P(Z<\\var{zupper})=1-\\var{p1} = \\var{prob2}$ to 2 decimal places.
\n2.
\n$\\simplify[all,!collectNumbers]{P({lower} < X < {upper}) = P(X < {upper})-P(X < {lower})}=P(Z<\\var{zupper})-P(Z<-\\var{zlower}) =\\var{p1}-\\var{p2} = \\var{prob3}$ to 2 decimal places.
", "statement": "The salary of an irish electrician is normally distributed with mean €{m} and standard deviation €{s}.
\n", "tags": ["checked2015", "MAS1403", "Rebel", "REBEL", "rebel", "rebelmaths"], "type": "question"}, {"name": "Inverse Normal Distribution (flour or Coke)", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Clodagh Carroll", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/384/"}, {"name": "Catherine Palmer", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/423/"}], "tags": [], "metadata": {"description": "\n
Given a random variable $X$ normally distributed as $\\operatorname{N}(m,\\sigma^2)$ find probabilities $P(X \\gt a),\\; a \\gt m;\\;\\;P(X \\lt b),\\;b \\lt m$.
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "Let $X$ represent the {chosen_amount} of {chosen_stuff}. It is known that $X$ follows a Normal distribution with a mean of {chosen_m}{chosen_units} and a standard deviation of {chosen_s}{chosen_units}.
\n", "advice": "\n
If {percentage}% of {chosen_item}s contain less than {x} {chosen_units} then we can write $P(X < x ) = \\var{prob}$
\nUsing the Standard Normal tables we can write $P(Z < \\var{z1}) = \\var{prob}$
\nNow $x = \\mu + z \\times \\sigma$
\nso that $x = \\var{chosen_m} + (\\var{z1}\\times \\var{chosen_s})= \\var{x1}$
\n", "rulesets": {}, "variables": {"prob1": {"name": "prob1", "group": "Ungrouped variables", "definition": "precround(1-p,2)", "description": "", "templateType": "anything"}, "Index": {"name": "Index", "group": "Ungrouped variables", "definition": "random(0,1)", "description": "", "templateType": "anything"}, "prob3": {"name": "prob3", "group": "Ungrouped variables", "definition": "precround(p1-p2,2)", "description": "", "templateType": "anything"}, "s": {"name": "s", "group": "Ungrouped variables", "definition": "[random(10..40#5), random(6..15#2)]", "description": "", "templateType": "anything"}, "item": {"name": "item", "group": "Ungrouped variables", "definition": "[\"packet\",\"bottle\"]", "description": "", "templateType": "anything"}, "x1": {"name": "x1", "group": "Ungrouped variables", "definition": "precround(normalinv(percentage/100, chosen_m,chosen_s),1)", "description": "", "templateType": "anything"}, "percentage": {"name": "percentage", "group": "Ungrouped variables", "definition": "random(1..10#1)", "description": "", "templateType": "anything"}, "chosen_m": {"name": "chosen_m", "group": "Ungrouped variables", "definition": "m[Index]", "description": "", "templateType": "anything"}, "m": {"name": "m", "group": "Ungrouped variables", "definition": "[random(950..1100#10),random(300..1000#50)]", "description": "", "templateType": "anything"}, "p2": {"name": "p2", "group": "Ungrouped variables", "definition": "1-p", "description": "", "templateType": "anything"}, "prob2": {"name": "prob2", "group": "Ungrouped variables", "definition": "precround(1-p1,2)", "description": "", "templateType": "anything"}, "chosen_amount": {"name": "chosen_amount", "group": "Ungrouped variables", "definition": "amount[Index]", "description": "", "templateType": "anything"}, "zupper": {"name": "zupper", "group": "Ungrouped variables", "definition": "precround((upper-chosen_m)/chosen_s,2)", "description": "", "templateType": "anything"}, "stuff": {"name": "stuff", "group": 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"item[Index]", "description": "", "templateType": "anything"}, "prob": {"name": "prob", "group": "Ungrouped variables", "definition": "percentage/100", "description": "", "templateType": "anything"}, "p": {"name": "p", "group": "Ungrouped variables", "definition": "precround(normalcdf(zlower,0,1),4)", "description": "", "templateType": "anything"}, "tol": {"name": "tol", "group": "Ungrouped variables", "definition": "0.01", "description": "", "templateType": "anything"}, "lower": {"name": "lower", "group": "Ungrouped variables", "definition": "random(chosen_m-1.5*chosen_s..chosen_m-0.5*chosen_s#5)", "description": "", "templateType": "anything"}, "upper": {"name": "upper", "group": "Ungrouped variables", "definition": "random(chosen_m+0.5*chosen_s..chosen_m+1.5*chosen_s#5)", "description": "", "templateType": "anything"}, "p1": {"name": "p1", "group": "Ungrouped variables", "definition": "precround(normalcdf(zupper,0,1),4)", "description": "", "templateType": "anything"}, "z1": {"name": "z1", "group": "Ungrouped variables", "definition": "precround(normalinv(percentage/100,0,1),2)", "description": "", "templateType": "anything"}, "chosen_units": {"name": "chosen_units", "group": "Ungrouped variables", "definition": "units1[Index]", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["units1", "upper", "lower", "p1", "m", "s", "zupper", "p", "amount", "p2", "tol", "zlower", "stuff", "prob2", "prob3", "prob1", "Index", "chosen_amount", "chosen_stuff", "chosen_units", "chosen_m", "chosen_s", "chosen_item", "item", "percentage", "x1", "z1", "prob"], "variable_groups": [{"name": "Unnamed group", "variables": []}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "
{percentage}% of {chosen_item}s contain less than {x} {chosen_units}.
Calculate the value of {x}.
{x} = [[0]] (rounded to the nearest {chosen_units})
", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "x1-0.5", "maxValue": "x1+0.5", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}]}, {"name": "Confidence Interval", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": [""], "variable_overrides": [[]], "questions": [{"name": "Cillian's copy of Find z-score for sample and calculate confidence interval", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Cillian Williamson", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/10206/"}], "tags": [], "metadata": {"description": "\n\t\tGiven mean and sd of 1000 sample returns on a scale of 1 to 7 together with a given score, find the z-score.
\n\t\tAlso find the 95% confidence interval for the population mean.
\n\t\t", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "\n\tA recent survey asked $\\var{samplesize}$ {these} to rate {this} on a scale from $\\var{bottom}$ ({expb}) to $\\var{top}$ ({expt}).
\n\tThe mean rating was $\\var{samplemean}$ with SD $\\var{sstdev}$.
\n\t \n\tEnter all values to 3 decimal places.
\n\t \n\t", "advice": "\n\ta)
\n\tThe $z$-score is given by
\n\t\\[z=\\frac{\\var{score}-\\var{samplemean}}{\\var{sstdev}}=\\var{zscore}\\]
\n\t(To 3 decimal places).
\n\tb)
\n\tThe lower bound for the 95% confidence interval is given by:
\n\tLower bound = $\\displaystyle \\var{samplemean}-1.96 \\times \\frac{ \\var{sstdev}}{\\sqrt{\\var{samplesize}}}=\\var{lowerbound}$
\n\t \n\tUpper bound = $\\displaystyle \\var{samplemean}+1.96 \\times \\frac{ \\var{sstdev}}{\\sqrt{\\var{samplesize}}}=\\var{upperbound}$
\n\t(Both to 3 decimal places.)
\n\tHence for the population mean $\\mu$ we can say that $\\var{lowerbound} \\le\\mu \\le \\var{upperbound}$ with $95$% confidence.
\n\t", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"expb": {"name": "expb", "group": "Ungrouped variables", "definition": "'Not at all important'", "description": "", "templateType": "anything", "can_override": false}, "top": {"name": "top", "group": "Ungrouped variables", "definition": "7", "description": "", "templateType": "anything", "can_override": false}, "score": {"name": "score", "group": "Ungrouped variables", "definition": "random(2..5#0.1)", "description": "", "templateType": "anything", "can_override": false}, "sstdev": {"name": "sstdev", "group": "Ungrouped variables", "definition": "random(0.9..2.0#0.01)", "description": "", "templateType": "anything", "can_override": false}, "upperbound": {"name": "upperbound", "group": "Ungrouped variables", "definition": "precround(samplemean+1.96*sstdev/sqrt(samplesize),3)", "description": "", "templateType": "anything", "can_override": false}, "these": {"name": "these", "group": "Ungrouped variables", "definition": "'UK shoppers'", "description": "", "templateType": "anything", "can_override": false}, "zscore": {"name": "zscore", "group": "Ungrouped variables", "definition": "precround((score-samplemean)/sstdev,3)", "description": "", "templateType": "anything", "can_override": false}, "this": {"name": "this", "group": "Ungrouped variables", "definition": "'the importance of price when making food choice decisions'", "description": "", "templateType": "anything", "can_override": false}, "expt": {"name": "expt", "group": "Ungrouped variables", "definition": "'Extremely important'", "description": "", "templateType": "anything", "can_override": false}, "samplemean": {"name": "samplemean", "group": "Ungrouped variables", "definition": "random(4.5..6.5#0.01)", "description": "", "templateType": "anything", "can_override": false}, "samplesize": {"name": "samplesize", "group": "Ungrouped variables", "definition": "1000", "description": "", "templateType": "anything", "can_override": false}, "bottom": {"name": "bottom", "group": "Ungrouped variables", "definition": "1", "description": "", "templateType": "anything", "can_override": false}, "lowerbound": {"name": "lowerbound", "group": "Ungrouped variables", "definition": "precround(samplemean-1.96*sstdev/sqrt(samplesize),3)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["zscore", "lowerbound", "bottom", "this", "top", "upperbound", "samplemean", "these", "sstdev", "score", "samplesize", "expb", "expt"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "10", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "\n\t\t\tWhat is the $z$-score for a score of $\\var{score}$?
\n\t\t\t \n\t\t\tEnter your answer to 3 decimal places.
\n\t\t\t", "minValue": "zscore-0.001", "maxValue": "zscore+0.001", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "\n\t\t\tCalculate the $95$% confidence interval for the population mean $\\mu$:
\n\t\t\tLower bound: [[0]]
\n\t\t\tUpper bound: [[1]]
\n\t\t\t \n\t\t\tEnter your answers to 3 decimal places.
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"}, "timedwarning": {"action": "warn", "message": "5 minutes left
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